On the deficiency of bipartite graphs

Giaro, Krzysztof; Kubale, Marek; Małafiejski, Michał

doi:10.1016/s0166-218x(99)00021-9

Cited by 41 publications

(55 citation statements)

References 4 publications

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“…For a vertex v 2 X 1 the arc colored 4 has been recolored 1, forming the interval [1,3]; analogously, we have the interval [3,5] for a vertex v 2 X 2 . Finally, the interval of the used colors for every vertex v i 2 A 0 is [4,6]. The Theorem 1.1 is proved.…”

Section: The Main Resultsmentioning

confidence: 91%

“…It has n ¼ 28 vertices and maximum degree Á ¼ 21. Later, other examples were found by Erdo }s (n ¼ 27 and Á ¼ 13), by Hertz and de Werra (n ¼ 21 and Á ¼ 14) [unpublished results], and by M. Malafiejski (n ¼ 19 and Á ¼ 15) [6]. The details on the first graph can be found in Refs.…”

Section: Introductionmentioning

confidence: 76%

“…f1; 2; 3; [3,6]. If u 2 U 1 then the arc colored 4 in the coloring g has been recolored 1 and another arc colored 4 has been added.…”

Section: The Main Resultsmentioning

confidence: 99%

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Interval coloring of (3,4)‐biregular bipartite graphs having large cubic subgraphs

Pyatkin

2004

Journal of Graph Theory

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An interval coloring of a graph is a proper edge coloring such that the set of used colors at every vertex is an interval of integers. Generally, it is an NP-hard problem to decide whether a graph has an interval coloring or not. A bipartite graph G ¼ (A,B;E) is (,)-biregular if each vertex in A has degree and each vertex in B has degree . In this paper we prove that if the (3,4)-biregular graph G has a cubic subgraph covering the set B then G has an interval coloring. ß

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Section: The Main Resultsmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 76%

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Interval coloring of (3,4)‐biregular bipartite graphs having large cubic subgraphs

Pyatkin

2004

Journal of Graph Theory

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“…In [7] it was observed that the both parts of an interval non-colorable bipartite graph should be relatively large.…”

Section: Related Workmentioning

confidence: 99%

“…Another example of interval non-colorable bipartite graph F on 19 vertices ( Fig. 1) was discovered by Mirumyan in 1989, but was not published, and was independently found by Giaro, Kubale and Malafiejski in [7]. On the other hand, in [6] it was shown that all bipartite graphs on at most 14 vertices are interval colorable.…”

Section: Introductionmentioning

confidence: 99%

On interval edge-colorings of bipartite graphs

Khachatrian¹,

Mamikonyan²

2015

2015 Computer Science and Information Technologies (CSIT)

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An edge-coloring of a graph G with colors 1, . . . , t is an interval t-coloring if all colors are used, and the colors of edges incident to each vertex of G are distinct and form an interval of integers. A graph G is interval colorable if it has an interval t-coloring for some positive integer t. The problem of deciding whether a bipartite graph is interval colorable is NP-complete. The smallest known examples of interval non-colorable bipartite graphs have 19 vertices. On the other hand it is known that the bipartite graphs on at most 14 vertices are interval colorable. In this work we observe that several classes of bipartite graphs of small order have an interval coloring. In particular, we show that all bipartite graphs on 15 vertices are interval colorable.

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Interval Non‐edge‐Colorable Bipartite Graphs and Multigraphs

Khachatrian

2013

Journal of Graph Theory

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An edge‐coloring of a graph G with colors 1,...,t is called an interval t‐coloring if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. In 1991, Erdős constructed a bipartite graph with 27 vertices and maximum degree 13 that has no interval coloring. Erdős's counterexample is the smallest (in a sense of maximum degree) known bipartite graph that is not interval colorable. On the other hand, in 1992, Hansen showed that all bipartite graphs with maximum degree at most 3 have an interval coloring. In this article, we give some methods for constructing of interval non‐edge‐colorable bipartite graphs. In particular, by these methods, we construct three bipartite graphs that have no interval coloring, contain 20, 19, 21 vertices and have maximum degree 11, 12, 13, respectively. This partially answers a question that arose in [T.R. Jensen, B. Toft, Graph coloring problems, Wiley Interscience Series in Discrete Mathematics and Optimization, 1995, p. 204]. We also consider similar problems for bipartite multigraphs.

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On the deficiency of bipartite graphs

Cited by 41 publications

References 4 publications

Interval coloring of (3,4)‐biregular bipartite graphs having large cubic subgraphs

Interval coloring of (3,4)‐biregular bipartite graphs having large cubic subgraphs

On interval edge-colorings of bipartite graphs

Interval Non‐edge‐Colorable Bipartite Graphs and Multigraphs

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